Choose the incorrect statement about the wo circles whose equations are given below x²+y²-10 x-10 y+41=0 and x²+y²-16 x-10 y+80=0 [17 Mar 2021 Shift 1]

Correct Answer is : Both circles' centres lie inside region of one another.

Distance between two centres is the average of radii of both the circles.

Both circles' centres lie inside region of one another.

Both circles pass through the centre of each other.

Circles have two intersection points.

Correct Answer is : Both circles' centres lie inside region of one another.

Test Index

Circles Part 1

Section: Mathematics

Question : 21 of 100

Marks: +1, -0

x^{2}+y^{2}-10 x-10 y+41=0

x^{2}+y^{2}-16 x-10 y+80=0

Solution:

C_{1} \Rightarrow x^{2}+y^{2}-10 x-10 y+41=0

(x-5)^{2}+(y-5)^{2}+41=25+25

\Rightarrow

Centre

=(5,5)

and radius

=3

\Rightarrow \; (x-5)^{2}+(y-5)^{2}=3^{2}

C_{2} \Rightarrow x^{2}+y^{2}-16 x-10 y+80=0

\Rightarrow \; (x-8)^{2}+(y-5)^{2}+80=64+25

\Rightarrow \; (x-8)^{2}+(y-5)^{2}=3^{2}

\Rightarrow

Centre

=(8,5)

and radius

=3

Now, distance between centres

=\sqrt{(8-5)^{2}+(5-5)^{2}}=3

Average radii

=\frac{3+3}{2}=3

\therefore

Option (a) is correct.

C_{1}(8,5)=(8-5)^{2}+(5-5)^{2}-9=0

C_{2}(5,5)=(5-8)^{2}+(5-5)^{2}-9=0

Centers of each other lies on circumference of each other.

Hence, (b) is incorrect statement.

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